Introductionto Binomial Theorem
In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums.
The Pascal's Triangle allows you to find the coefficients of the expanded form of the expression only if you write down the previous rows. Thus, this is very impractical when the value of the power n is large.
The binomial theorem makes use of the factorial notation (ie "!")
This is the function for finding the coefficient of the term in the expression.
Where "n" is the power of the factorised form [ ie. (x+y)^n ]
And "k" is the power of x in the term.
We know that the power of x and y in a term will add up to the power of the factorised form.
Thus (n-k) is the power of y in the term.
The L.H.S of the equation (n and k in the brackets) represents the method of writing the term's n and k.
Thus to find, for example, the first coefficient of the expanded (x+y)^5:
We know that n=5
We also know that the first term of the expression has x with power of n and y with power of 0.
Thus the equation is 5! / 5! x 0! = 5! / 5! x 1 = 5! / 5! = 1
Thus the first coefficient is one, which matches with the Pascal's Triangle.
Another example:
Find the coefficient of the 3rd term of the expanded expression of (x+y)^6.
We know that the third term's x has a power "k" of n-(position of term - 1) = n-(3-1) = n-2 = 6-2 = 4
Thus, n is 6, k is 4 and (n-k) is 2.
Sub in these values into the R.H.S function:
And you get 6! / 4! x 2! = 6(5)(4)(3)(2)(1) / (4)(3)(2)(1) x (2)(1) = 15
When you plot a Pascal's Triangle or expand manually, you get the same answer.
Thus when you have mastered the skill of using the binomial theorem, you can expand binomial expressions quickly.
Road Junction
To recap on binomials, click on Binomials.
To move on to learn more about the Pascal's Triangle, click on Pascal's Triangle.
To explore about another kind of numerical geometrical arrangement, click on Leibniz Harmonic Triangle.
Continue discovering more about the Pascal's Triangle!
In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums.
The Pascal's Triangle allows you to find the coefficients of the expanded form of the expression only if you write down the previous rows. Thus, this is very impractical when the value of the power n is large.
The binomial theorem makes use of the factorial notation (ie "!")
For example:
1! = 1
2! = 2(1)
3! = 3(2)(1)
4! = 4(3)(2)(1)
5! = 5(4)(3)(2)(1)
This is the function for finding the coefficient of the term in the expression.
Where "n" is the power of the factorised form [ ie. (x+y)^n ]
And "k" is the power of x in the term.
We know that the power of x and y in a term will add up to the power of the factorised form.
Thus (n-k) is the power of y in the term.
The L.H.S of the equation (n and k in the brackets) represents the method of writing the term's n and k.
Thus to find, for example, the first coefficient of the expanded (x+y)^5:
We know that n=5
We also know that the first term of the expression has x with power of n and y with power of 0.
Thus the equation is 5! / 5! x 0! = 5! / 5! x 1 = 5! / 5! = 1
Thus the first coefficient is one, which matches with the Pascal's Triangle.
Another example:
Find the coefficient of the 3rd term of the expanded expression of (x+y)^6.
We know that the third term's x has a power "k" of n-(position of term - 1) = n-(3-1) = n-2 = 6-2 = 4
Thus, n is 6, k is 4 and (n-k) is 2.
Sub in these values into the R.H.S function:
And you get 6! / 4! x 2! = 6(5)(4)(3)(2)(1) / (4)(3)(2)(1) x (2)(1) = 15
When you plot a Pascal's Triangle or expand manually, you get the same answer.
Thus when you have mastered the skill of using the binomial theorem, you can expand binomial expressions quickly.
Road Junction
To recap on binomials, click on Binomials.
To move on to learn more about the Pascal's Triangle, click on Pascal's Triangle.
To explore about another kind of numerical geometrical arrangement, click on Leibniz Harmonic Triangle.
Continue discovering more about the Pascal's Triangle!